All graphs in this paper are finite and simple. For general graph theory notation which are not defined here we follow [1]}.
Let \(P_n\) , \(C_n\)  and \(K_n\)  denote the path, cycle and complete graph on \(n\)  vertices, respectively.
We denote the complement graph of \(G\)  by \(\overline{G}\) . For two vertex disjoint graphs \(G\)  and \(H\) , we write \(G+H\)  to denote the disjoint union of \(G\)  and \(H\) , and \(G\vee H\) 
to denote the graph obtained from \(G+H\)  by adding an edge between every vertex in \(G\)  and every vertex in \(H\) . For a positive integer \(r\) , we use \(rG\)  to denote the disjoint union of \(r\)  copies of \(G\) . A hole is an induced cycle on four or more vertices. An antihole is the complement graph of a hole. A hole or antihole is odd (even) if it has an odd (even) number of vertices. A clique (stable set) in a graph \(G\)  is a set of pairwise adjacent (nonadjacent) vertices in \(G\) .
We say that a graph \(G\)  contains a graph \(H\)  if \(G\)  has an induced subgraph that is isomorphic to \(H\) . A graph \(G\)  is \(H\) -free if it does not contain \(H\) . For a family \(\mathcal {H}\)  of graphs, \(G\)  is \(\mathcal {H}\) -free if \(G\)  is \(H\) -free for every \(H\in \mathcal {H}\) . We write \((H_1,\ldots ,H_n)\) -free instead of \(\lbrace H_1,\ldots ,H_n\rbrace \) -free.
